Exponential sum working group


Understand Bourgain-Chang's proof of bounds on exponential sums over small multiplicative subgroups.


As a starting point, we will use

Bourgain, J.(1-IASP); Chang, M.-C.(1-CAR) Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z\sb Q\sp *$, where $Q$ is composite with few prime factors. Geom. Funct. Anal. 16 (2006), no. 2, 327--366.

(Or click here.)

First meeting

Speaker: Pär Kurlberg

Time/place: friday 070223, 14.15-15.15, room 3733.

Topics: Some classical bounds on complete exponential sums, such as Gauss and Kloosterman sums, the Weil bounds (aka the Riemann hypothesis for curves). Bounds on incomplete sums (Polya-Vinogradov and Burgess), and some applications (e.g. the size of the smallest quadratic non-residue.)

Second meeting

Speaker: Pär Kurlberg

Time/place: tuesday 070227, 10.15-12, room 3721.

Topic: We will start with an overview of the proof.

Third meeting

Speaker: Pär Kurlberg

Time/place: tuesday 070306, 10.15-12, room 3721.

Topic: We will wrap up the sketch of the proof by finishing the "statistical" multiplicative invariance. We will then go onto the proof of proposition 2.1.

Fourth meeting

Time/place: tuesday 070313, 10.15-12, room 3721.

Speaker: Pär Kurlberg

Topic: We will finish the proof of proposition 2.1 - how to use BGS' (Balog-Gowers-Szemeredi) to go from "statistical" multiplicative invariance to true multiplicative invariance.

Fifth meeting

Time/place: wednesday 070314, 13.00-15.00, room 3733.

Speaker: Alexander Engström

Topic: On addititive combinatorics and the proof of Balog-Gowers-Szemeredi.

Sixth Meeting

Time/place: 10:15 - 12:00, Tuesday 20 March 2007, room 3721.

Speaker: Liangyi Zhao

Topic: I'll talk about bounds for character sums due to D. A. Burgess. In the interest of time, I will not go through all the details of the proof but present only the key points. Also, I will simply quote some results of H. Hasse and A. Weil which are needed in the proof.


D. A. Burgess, On Characters Sums and Primitive Roots, Proc. London Math. Soc., No. 3, Vol. 12, 1962, pp. 179 - 192

Lecture notes